I am trying to prepare for my final exam in NT. Any help would be appreciated. These questions are just for study purposes, I messed them up on my homework, etc.
Reduced Residue System Proof
Show that if a=b(mod n) then (a,n)= (b,n)
I am trying to prepare for my final exam in NT. Any help would be appreciated. These questions are just for study purposes, I messed them up on my homework, etc.
Reduced Residue System Proof
Show that if a=b(mod n) then (a,n)= (b,n)
(mod n) for some integer k
Now we show the set of divisors of a and n are the same as the set of divisors of b and n.
Let d divide a and n.
then d divides the RHS so it divides b as well.
Let d divide b and n.
then d divides the LHS so it divides a as well.
Thus the set of divisors is the same, so their greatest element is the same. so (a,n)=(b,n)
Maybe what you meant was being prime implies for . If so, I would check out this:Fermat number - Wikipedia, the free encyclopedia
For completeness, it should state the n could also be 0.
The statement should be
IF is prime, then n is a power of 2 or n=0.
the proof is given here Fermat number - Wikipedia, the free encyclopedia
There is nothing complicated about the proof supplied in the link. It is a proven fact, this has to be what the teacher was going for, the other possibilities are either not true or would never be asked on an exam by someone with a PhD in math.
I am not following what your problem is with this, it is a very straightforward if then statement. The proof goes by contradiction in supposing that n is not a power of two, then it has an odd prime factor. Then you show how you factor which makes it NOT prime, a contradiction.